Heuristics on Tate Shafarevitch Groups of Elliptic Curves Defined over Q

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1 Heuristics on Tate Shafarevitch Grous of Ellitic Curves Defined over Q Christohe Delaunay CONTENTS. Introduction 2. Dirichlet Series and Averages 3. Heuristics on Tate Shafarevitch Grous References In a well-known aer, Cohen and Lenstra gave conjectures on class grous of number fields. We give here similar conjectures for Tate Shafarevitch grous of ellitic curves defined over Q. For such grous (if they are finite), there exists a nondegenerate, alternating, bilinear airing. We give some roerties of such grous and then formulate heuristics which allow us to give recise conjectures.. INTRODUCTION We make a study of Tate{Shafarevitch grous of ellitic curves dened over Q similar to the one made in [Cohen and Lenstra 984] of class grous of number elds. art of our motivation is the dee analogy that exists between these grous. In this aer, we will assume the truth of the conjecture asserting that the Tate{Shafarevitch grou of an ellitic curve over Q is nite. Under this conjecture, there exists a nondegenerate, alternating, bilinear airing! Q =Z see [Silverman 986], for examle. We will say that a air (G ) is a grou of tye S if G is a nite abelian grou and is a nondegenerate alternating bilinear airing GG! Q =Z. We will also have to consider isomorhism classes of grous of tye S, where two grous (G ) and (G 2 2 ) of tye S are said to be isomorhic if there exists an isomorhism G! G 2 such that 2 ((x) (y)) = (x y) for all x y 2 G. In [Cohen and Lenstra 984], the grous considered are simly nite abelian grous, and the main idea is to give each grou G a weight roortional to =jaut Gj. Here we must relace Aut G by its analog Aut s G, the grou of automorhisms of (G ) that reserve. c A K eters, Ltd /200 $0.50 er age Exerimental Mathematics 02, age 9

2 92 Exerimental Mathematics, Vol. 0 (200), No. 2 We will use the following notation is as an abbreviation for a sum over all isomorhism classes of grous (G ) of tye S of order n 2. We denote by the set of rime numbers and, for 2, we denote by r (G) the -rank of G. If (G ) is a grou of tye S, we dene = jaut s Gj w s a(g) = jaut s Gj jjgj (= 2 ) a (= 2 ) a r(g)=2 where (q) a = Q ia ( qi ) for a 2 N [ fg. Finally, we set w s (n 2 ) = ws (G) and w s a(n 2 ) = ws a(g), and note that w s a(g) tends to when a tends to innity. We give some roerties of grous of tye S. We start with an examle. Let G = (Z = a Z Z = a Z ) (Z = a 2 Z Z = a 2 Z ) (Z = a j Z Z = a j Z ) with a a 2 a j. Denote by e e 2 e 2j a \canonical basis" (for examle the i-th comonent of e i is taken to be invertible mod, and the others are taken equal to zero). Dene on this basis by (e 2i e 2i ) = (e 2i e 2i ) = = a i 2 Q =Z (e i e j ) = 0 elsewhere. Then (G ) is a grou of tye S. Hence, if a nite grou G is isomorhic to H H for a suitable grou H, we can dene, as above, a nondegenerate, alternating, bilinear airing (the -comonents of (G ) are orthogonal). We now show the converse. Lemma. Let (G ) be a -grou of tye S, and let x 2 G be an element of maximal order, say k. Then There exists y 2 G of order k with (x y) = = k 2 Q =Z. There exists a subgrou H of G with (H j HH ) of tye S, such that G = (hxi hyi)? H, where? denotes an orthogonal direct sum. roof. The element y exists because is nondegenerate. We can then set H = fz 2 G (x z) = (y z) = 0g By induction on the order of the grou, this lemma allows us to rove the following roosition. roosition 2. If (G ) is a -grou of tye S, then (G ) is isomorhic to the grou of the examle given above, for aroriate (a i ). In articular, G ' H H and the structure of Aut s G is indeendent of. Recall that abelian grous of order n are in oneto-one corresondence with artitions of n. If () is a artition of n, we denote by i the number of occurrences of i in (). Clearly n = j j where j is the largest integer with j 6= 0. Further, we denote by 2 j the arts of the associated artition of n dened by k = k + k+ + + j. Clearly n = j. Theorem 3. Let (G ) be a grou of tye S of order 2n with G ' H H and H = (Z =) Z = 2 2 Z = j j Then, with the receding notation, jaut s Gj = 2(2 ++2 j )+n 2 ij roof. We reason by induction on n. Let e e 2 be a \canonical basis" with e, e 2 of order j. Take e and e 2 to be the elements of G with on the last and enultimate comonents, resectively, and zero elsewhere. Let be the airing of the examle. An automorhism g is given by the image of a basis. Let x be the image of e. There exist 2n ( = 2 j ) ossibilities for x, this being the number of elements of order j in G. If z is the image of e 2, we must have (x z) = = j. Writing G = (hxi hyi)? H we deduce that there are 2n j ossibilities for z = g(e 2 ). Finally, note that e 3 e 4 form a \canonical basis" of a grou of tye S which is isomorhic to (H jhh ) and that g(e 3 ) g(e 4 ) must belong to H. So there are jaut s Hj ossibilities we then use our induction hyothesis since jhj < jgj. Remark. Taking = n, we obtain the order of the symlectic grou S (2n ). i

3 Delaunay Heuristics on Tate Shafarevitch Grous of Ellitic Curves Defined over Q DIRICHLET SERIES AND AVERAGES Study of w s a (n2 ) and w s (n 2 ) Theorem 4. G s ( 2n ) w s a(g) = 3n (= 2 ) n+a (= 2 ) a (= 2 ) n roof. In [Cohen and Lenstra 984] we nd the formula (=) a = (=) n+a jaut Gj (=) a r (G) n (=) a (=) n G( n ) where the sum is over all abelian grous of order n u to isomorhism. Furthermore, it is well-known (and can be roved as Theorem 3) that for a -grou G corresonding to () we have jaut Gj = j see [Hall 938]. We deduce that (=) n (=) a r() ()=n (=) j (=) (=) j ij i = n (=) n+a (=) a (=) n where the sum is over all artitions () of n and r() is the number of arts in (). The above formula is in fact a formal identity. It follows that we can substitute 2 for and multily by = n, roving the theorem. Corollary 5. w s a(g) = n 3 kn Letting a tend to, we obtain Corollary 6. Aut s (G) = n 3 (= 2 ) +a (= 2 ) a (= 2 ) kn (= 2 ) Definition 7. We dene the functions s a and s by the Dirichlet series s (z) = n= w s (n) n z s a(z) = n= w s a(n) n z with w s (n) = w s a(n) = 0 if n is not a erfect square. The next lemma can be roved by induction on a Lemma 8. Let a be a nonnegative integer. Then () +a () () a = a+ j= j Remark. Letting a tend to, we obtain a formula of Euler Theorem 9. We have s a(z) = s (z) = () = n= n= w s a(n) n z = w s (n) n z = j= j a j= j= (2z+2j +) (2z+2j +) roof. It is sucient to rove the rst formula, the second following by letting a!. We rst write s a(z) = Q 2z w s a( 2 ) and relace w s a( 2 ) by the formula of Theorem 4. We obtain s a(z) = and we use Lemma 8 to conclude. Averages (= 2 ) +a 2 (= 2 ) a (= 2 ) (2z+) Let f be a comlex-valued function dened on isomorhism classes of grous of tye S. Definition 0. Dene w s (f n 2 ) = f(g) w s (f n) s (f z) = n z n wa(f s n 2 ) = wa(g)f(g) s s a(f z) = n w s a(f n) n z with w s (f n) = w s a(f n) = 0 if n is not a erfect square. When u 0, dene c au (f n) by n= c au (f n) n z = s a(f z+u)a(z) s a(z+u) s Definition. The (a u)-average of f is dened by M s au(f) = lim x! nc nx au(f n) nx nws a(n) If a =, we will seak of the u-average of f, and we will write M s u(f) instead of M s u(f).

4 94 Exerimental Mathematics, Vol. 0 (200), No. 2 Remarks. This denition is analogous to that of the (a u)-average in [Cohen and Lenstra 984, Denition 5. and roosition 5.4]. In articular, the (a u)-average of f, if it exists, is an average (that is, the (a u)-average of a constant function is equal to that constant). If f is the characteristic function of a roerty, we will seak of the (a u)-robability of or simly of the u-robability of. Imortant we have included the factor n in the denition so that the denominator diverges as x!. Moreover, we can see that the average of f is unchanged (for a reasonable class of functions) if we relace n by n l for l. This is not true if we relace n by n l for l <, in articular if we relace n by the constant. We also need the following Tauberian theorem roosition 2 [Tenenbaum 995]. Let (c(n)) n be nonnegative real numbers. If D(z) = n c(n)=nz converges for Re z > 0 and if D(z) C=z can be analytically continued to Re z 0, then nx c(n) C log x Alying this to a(z ) s (i.e., to c(n) = nwa(n)), s we obtain Q nwa(n) s 2ka (2k ) log x 2 nx The Tauberian theorem immediately imlies roosition 3. Let f be a nonnegative function de- ned on isomorhism classes of grous of tye S. If s a(f z ) converges for Re z > 0 and s a(f z ) C=z can be analytically continued to Re z 0, then. For u = 0, M s a0(f) = Q2ka 2C (2k ) = lim z!0 2. For u 6= 0, M s au(f) = s a(f u ) s a(u ) s a(f z ) s a(z ) For our alications, we need to be able to restrict our attention to -arts of grous of tye S, where is a set of rime numbers. For this, we denote by f the function G 7! f(g ), where G is the -art of G. We also write N = fn 2 N jn =) 2 g. roosition 4. Let be a set of rime numbers. Then w s a(f n) = w s a(f n )w s a(n 2 ) where n = n n 2 and n is the -art of n. In articular, a(f s z) equals a(f n) ws n z n2n 62 a k= (=) 2z+2k+ roof. Immediate consequence of the denitions. We now give some examles of averages, with follow from roositions 3 and 4. For simlicity we assume a =. Examle A. Let 2 R and u >. The u-average of jgj is equal to Q (2u 2 + 2j ) j= Q (2u + 2j ) j= In articular, if u 2, the u-average of jgj equals (2u ). Examle B. Let L be a grou of tye S with L a - grou. The u-robability that the -art of a grou of tye S is isomorhic to L is equal to jlj u jaut s Lj k= 2u+2k Examle C. Assume that all rime divisors of n are in. The u-robability that the -art of a grou of tye S has cardinality equal to n is equal to n u w s (n) k= 2u+2k Examle D. The u-robability that G 6= f0g is equal to (=) 2u+2k k= Examle E. The u-robability that the -art of a grou of tye S is isomorhic to the square of a cyclic grou is = 2 + = 2u+3 ( = 2 ) (=) 2u+2k k=2 In articular, when = this robability equals (2) ( = 2 += 2u+3 ) (2u+3)(2u+5)(2u+7)

5 Delaunay Heuristics on Tate Shafarevitch Grous of Ellitic Curves Defined over Q 95 Averages Involving -Ranks We can also obtain results on the -rank of a grou of tye S. For simlicity, we assume a = but the results can also be given for nite values of a. roosition 5. Let and r be two nonnegative integers with r. We have G s ( 2 ) r (G)=2r = w s ( 2 ) (= 2 ) (= 2 ) 2r 2 +2r (= 2 ) r (= 2 ) r (= 2 ) r roof. We use exactly the same methods as in the roof of Theorem 4, using the formula of [Cohen and Lenstra 984, Theorem 6.]. Corollary 6. Let n 2 Z with kn. Then r (G)=2r = w s (n 2 ) (= 2 ) (= 2 ) 2r 2 +2r (= 2 ) r (= 2 ) r (= 2 ) r roof. Write n = n 2 and use the multilicativity of w s. roosition 7. Let r be a nonnegative integer. Then G s (2r) jgj z = r(2r+2z+) (= 2 ) r r j= = 2z+2j+ where the sum is over all -grous (G f) of tye S with r (G) = 2r. roof. We write G s (2r) jgj z = 2z G( 2 ) r (G)=2r and we obtain the formula by using roosition 5, Theorem 4 and Lemma 8. We thus obtain Examle F. The u-robability that the -rank of a grou of tye S is 2r equals r(2u+2r ) (= 2 ) r k=r+ = 2u+2k The next result is roved using [Cohen and Lenstra 984, Theorem 6.4] roosition 8. If are two nonnegative integers, then G s ( 2 ) 0i< r (G) 2i = ws ( 2 2 ) Corollary 9. If n is a nonnegative integer with jn, then 0i< This corollary gives r (G) 2i = ws (n 2 = 2 ) Examle G. The 0-average of the function r (G) is HEURISTICS ON TATE SHAFAREVITCH GROUS Using the analogy between units of number elds and rational oints on ellitic curves, we can now give a \Cohen{Lenstra"-tye heuristic assumtion for Tate{Shafarevitch grous of ellitic curves de- ned over Q, and deduce from them and the above results on grous of tye S a number of conjectures on Tate{Shafarevitch grous. Let E u be the set of isomorhism classes of ellitic curves E of rank u dened over Q, which we assume to be ordered by the conductor N(E). For a function f dened on isomorhism classes of grous of tye S, we dene! u (f x) = E2E u N(E)x f((e))! u (x) = E2E u N(E)x We can dene an average of f by setting! u (f x) M u (f) = lim x!! u (x) The basic heuristic assumtion is then the following Heuristic Assumtion. M u (f) = M s u=2 (f). We now give some consequences of this assumtion. The Rank-Zero Case The robability that is isomorhic to the square of a cyclic grou is (2) ( = 2 + = 3 ) (3)(5)(7)

6 96 Exerimental Mathematics, Vol. 0 (200), No. 2 or aroximately The robability that divides jj is f 0 () = (=) 2k k= In articular, f 0 (2) ' , f 0 (3) ' , and f 0 (5) ' We give here the robability that the -art of is isomorhic to a grou G G/ (Z =Z ) 2 (Z =Z Z =Z ) 2 (Z = 2 Z ) The robability that r () = 2r is r(2r ) (= 2 ) r The Rank-One Case k=r+ = 2k The robability that is isomorhic to the square of a cyclic grou is (2) k=4 (2k) The robability that divides jj is f () = (=) 2k k= In articular, f (2) ' 0346, f (3) ' 02344, and f (5) ' Let L be a grou of tye S. The robability that is isomorhic to L is jlj jaut s Lj (2)(4)(6) In articular, jj = with robability close to A roof of the heuristic assumtion (and of its consequences) is resently out of reach. It is also dicult to check numerically our conjectures, since nontrivial Tate{Shafarevitch grous seem to aear whenever the conductor is very large and tables of ellitic curves have been done \only" for N 6000 [Cremona 992]. Nevertheless, we make some comments that oint toward the truth of the heuristic assumtion above. First, we note that the nature of the results are dierent according to the arity of the rank (in the sense that they involve values of the Riemann zeta function at odd ositive integers or at even ositive integers). This seems quite natural since ellitic curves can be naturally slit into two arts according to the sign of the functional equation, in other words according to the arity of the rank if we assume the Birch and Swinnerton-Dyer conjecture. The conjectures also redict that the -rank of a Tate{Shafarevitch grou with a nontrivial -art is often equal to 2. Indeed, all the nontrivial -arts of Tate{Shafarevitch grous in Cremona's table have a -rank equal to 2 [Cremona and Mazur 2000]. REFERENCES [Cohen and Lenstra 984] H. Cohen and H. W. Lenstra, Jr., \Heuristics on class grous of number elds",. 33{62 in Number theory (Noordwijkerhout, 983), edited by H. Jager, Lecture notes in Math. 068, Sringer, Berlin, 984. [Cremona 992] J. E. Cremona, Algorithms for modular ellitic curves, Cambridge Univ. ress, Cambridge, 992. Second edition, 997. [Cremona and Mazur 2000] J. E. Cremona and B. Mazur, \Visualizing elements in the Shafarevich-Tate grou", Exeriment. Math. 9 (2000), 3{28. [Hall 938]. Hall, \A artition formula connected with Abelian grous", Comment. Math. Helv. (938), 26{29. [Silverman 986] J. H. Silverman, The arithmetic of ellitic curves, vol. 06, Graduate Texts in Math., Sringer, New ork, 986. [Tenenbaum 995] G. Tenenbaum, Introduction a la theorie analytique et robabiliste des nombres, 2nd ed., Cours secialises, Soc. math. France, aris, 995. Christohe Delaunay, Universite Bordeaux I, Laboratoire A2, 35 Cours de la Liberation, Talence, France (delaunay@math.u-bordeaux.fr) Received Aril 24, 2000 acceted in revised form October 0, 2000